Uniform boundedness conjecture for scattering states of bounded compactly supported potentials

Let $V$ obey $|V| \le V_0$, and $V(x)=0$ when $|x|\ge R_0$, for some finite $V_0,R_0$. Let $\hat H=\frac{\hat p^2}{2M}+V(\hat x)$, and let $\psi$ be any scattering state of $\hat H$.

Uniform boundedness conjecture. There is a universal constant $B_{V_0,R_0}$ such that

$$|\psi|\le B_{V_0,R_0},$$

where $B_{V_0,R_0}$ is independent of the details of $V$, though it may depend on the incident momentum and mass. This conjecture would provide a uniform $L_\infty$ bound needed for stability estimates across the class of bounded potentials with bounded support; the supplied text does not establish the claim or indicate a resolution.